| The broad theme of this dissertation is inventory management and procurement management. The dissertation includes three essays, whose summaries are presented in the paragraphs below. In the first essay, we study fixed-dimensional stochastic dynamic programs in a discrete setting over a finite horizon. Under the primary assumption that the cost-to-go functions are discrete L-natural-convex, we propose a pseudo-polynomial time approximation scheme that solves this problem to within an arbitrary pre-specified additive positive error. The main technique we develop for deriving our scheme is the approximation of a fixed-dimensional L-natural-convex function on a bounded rectangular set, using only a selected number of points in its domain. Our approximation scheme is illustrated on a well-known problem in inventory theory, namely the single-product problem with lost sales and lead times. In the second essay, we study several finite-horizon, discrete-time, dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi-period extension and a single-product, multi-echelon assembly model. For all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the single-product, multi-echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely rolling-horizon optimization. In the third essay, we consider the following problem: A firm is soliciting bids from a fixed-sized pool of yet-to-be-qualified suppliers for an indivisible contract. The contract can only be awarded to a supplier who passes a multi-stage qualification process. In each stage, the buyer selects a subset of those suppliers who have passed all previous stages and tests them simultaneously. The buyer incurs a fixed testing cost in each stage for each supplier she chooses to test. The buyer seeks an optimal mechanism, i.e., one that minimizes her total expected cost. Our main results are optimal mechanisms for (i) a symmetric setting, where suppliers' private costs follow the same distribution and the passing probability in a stage is the same for the suppliers, and (ii) an asymmetric setting, where there are two different pools of suppliers -- new and mature -- that differ in their costs and passing probabilities. |
